10000的阶乘的算法(大数的阶乘) [转]
很多天没有更新自己的Blog了,前几天拿到一个题目.就是写一段程序计算10000的阶乘.当时我还以为这题目非常简单,没想到还是需要动点大脑的.花了将近半个小时才搞定,拿出来分享一下.
为什么不能用普通的方法来写呢,比如说递归?在我的教科书上可是用的是递归呀?不知道你注意没有,如果是100的阶乘的话,其结果肯定是非常大的,以我们现有语言的数据类型肯定是没法使用的,就拿C来说,long型能存的下10000的阶乘吗?未必.所以我就使用数组来存储结果的每一位,然后输出每一位不就是结果吗.
那么具体怎样去做?
首先确定结果的位数?如何确定呢?请看下面. 2!=1*2<=10*10
3!=1*2*3<=10*10*10
.......
所以我们可以得出一个结论
n!<=10n
所以n!的位数可以这样计算:
两边取对数,即log10n!<=log1010n
两边n>=Log101+Log102+Log10 3+....Log10 n
这样n!的位数肯定等于小于Log101+Log102+Log10 3+....Log10 n.以上是错误的正确的推断如下:
可以将n!表示成10的次幂,即n!=10^M(10的M次方)则不小于M的最小整数就是 n!的位数,
对该式两边取对数,有=log10^n!即:
M = log10^1+log10^2+log10^3...+log10^n
循环求和,就能算得M值,该M是n!的精确位数。
位数的确定解决之后,就看看如何计算了.
看看如下代码:
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int index=0;
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long carrier=0;
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double bitCount = 1;
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int begin = 0;
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for(index=2; index<=n; ++index)
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{
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long multiValue = 0;
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bitCount += log10((long double)index);
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if(arrValue[begin] == 0)
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begin++;
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for(int j=begin; j<int(bitCount); ++j)
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{
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multiValue += (index*arrValue[j]);
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arrValue[j] = char(multiValue % 10);
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multiValue /= 10;
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}
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}
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这里就是计算的关键了.注意一下进位问题即可.所有代码如下:
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//////////////////////////////////////////////////////////////////////////
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// Date created: 2005/07/12
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// Author: Confach Zhang
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// Purpose: 计算n!的值
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//////////////////////////////////////////////////////////////////////////
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using namespace std;
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#include "StdAfx.h"
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#include <iostream.h>
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#include <conio.h>
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#include <stdlib.h>
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#include <math.h>
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#include <stdio.h>
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#include <iomanip.h>
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int GetNumber(); //输入 n
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int GetBitLength(int n); //求n!的位数
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char* Initialize(int); //初始化存储结果的值
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void PrintValue(char *a,int size); //打印值到屏幕
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void PrintValue(char *a,int size,char* fileName); //打印值到文件
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char* GetValue(int val); //计算
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char* SubGetValue(char* ,int);
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int main()
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{
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int value=GetNumber();
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char fileName[16];
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int size=GetBitLength(value);
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char *pa = Initialize(size);
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//pa=GetValue();
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pa=GetValue(value);
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PrintValue(pa,size);
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//sprintf(fileName,"%s","10000!.txt");
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sprintf(fileName,"%d!.txt",value);
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PrintValue(pa,size,fileName);
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delete []pa; //note:
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return 1;
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}
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//函数GetValue
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// 求得计算结果
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//返回结果
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//History:
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//1)char* GetValue()
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//2)GetValue(int val)
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// 参数:val 计算阶乘的值
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char* GetValue(int val)
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{
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//定义一个数组存储阶乘的值
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//首先得到10000!阶乘的位数
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int VALUE=val;
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int length=GetBitLength(VALUE);
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char *arrValue = new char[length];
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if(!arrValue) {
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cout <<"申请内存失败!" << endl;
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exit(1);
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}
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arrValue[0] = 1;
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for(int i=1; i<length; i++)
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arrValue[i] = 0;
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arrValue=SubGetValue(arrValue,VALUE);
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return arrValue;
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}
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char* SubGetValue(char* arrValue,int n)
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{
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int index=0;
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long carrier=0;
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double bitCount = 1;
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int begin = 0;
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for(index=2; index<=n; ++index)
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{
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long multiValue = 0;
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bitCount += log10((long double)index);
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if(arrValue[begin] == 0)
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begin++;
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for(int j=begin; j<int(bitCount); ++j)
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{
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multiValue += (index*arrValue[j]);
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arrValue[j] = char(multiValue % 10);
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multiValue /= 10;
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}
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}
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return arrValue;
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}
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//得到计算阶乘的值,此函数为新增
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int GetNumber()
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{
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int n;
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cout << "请输入要计算阶乘的n值: ";
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cin >> n;
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while(n < 0) {
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cout << "输入错误,请重新输入: ";
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cin >> n;
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}
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if(n == 0)
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exit(1);
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return n;
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}
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//函数GetBitLength
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// 求得计算结果的位数,本函数为新增加
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//参数
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// n 需要计算的阶乘的数
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//返回结果的位数
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int GetBitLength(int n)
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{
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double sum = 1.0;
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for(int i=1; i<=n; i++)
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sum += log10((long double)i);
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return int(sum);
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}
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//-----------
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//函数:Initialize
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// 初始化存储结果的数组
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//参数:
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// size 数组的长度
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//返回值
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// 初始化后的数组
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//-------------
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char * Initialize(int size)
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{
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char *arrValue = new char[size];
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if(!arrValue) {
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cout << size<<"太大,申请内存失败!" << endl;
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exit(1);
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}
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arrValue[0] = 1;
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for(int i=1; i<size; i++)
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arrValue[i] = 0;
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return arrValue;
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}
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//-----------
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//函数:PrintValue
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// 将结果输入到屏幕上
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//参数:
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// buff 存储结果的数组
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// buffLen 数组的长度
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// fileName 文件名
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//-------------
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void PrintValue(char *buff, int buffLen)
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{
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int bit = 0;
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int nCol=0;
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for(int i=buffLen-1; i>=0; i--) {
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if(bit % 10 == 0)
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{
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cout << " " ;
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nCol++;
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if(nCol==10)cout<<endl;
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}
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cout << int (buff[i]);
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bit++;
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}
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cout << endl;
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}
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//-----------
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//函数:PrintValue
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// 将结果输入到一个文件中
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//参数:
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// buff 存储结果的数组
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// buffLen 数组的长度
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// fileName 文件名
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//-------------
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void PrintValue(char *buff,int buffLen,char *fileName)
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{
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int bit = 0;
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int nCol=0;
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FILE *fp=NULL;
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//-----------------------------
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if (fileName==NULL) return ;
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fp=fopen(fileName,"wt");
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if (fp==NULL)
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{
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printf("不能创建文件%s",fileName);
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return ;
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}
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for(int i=buffLen-1; i>=0; i--)
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{
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fprintf(fp,"%d",int(buff[i]));
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if(bit % 9 == 0)
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{
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fprintf(fp,"%s"," ");
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nCol++;
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if(nCol==8)
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{
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fprintf(fp,"%s","\n");
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nCol=0;
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}
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}
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bit++;
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}
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fprintf(fp,"\n");
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fclose(fp);
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}
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